Computing knot/link groups The knot group of a knot $K$ is the fundamental group of $\mathbb R^3 \smallsetminus K$; that is, the set of possibly self-crossing closed paths (starting and ending at any single point in space) which you can take through real-space which are not equivalent to each other if you forbid passing through the locus of the knot; and where you can compose the paths in the natural way. For instance, the knot group of the un-knot is just $\mathbb Z$, corresponding to the number of times that a path winds about a cycle in space.
We can consider similar questions for links ("knots" having more than one closed loop) as for knots of a single component: given a link, we can consider the group of closed paths in space which avoids crossing either of the two components.
I have a friend who has a tattoo of a knot (more precisely, a link of two components) in the shape of a maple leaf; we were curious what properties it had. A presentation of the link is given below. How would one obtain a presentation of the link group? Does the link group of this particular link correspond to any relatively nice group?
 
 A: I plugged this into SnapPy and got the following presentation: 
Generators: a,b,c,d
Relators: abCBACADadcDAdabCBADadCDAdadcDAdabcBADadCDAdacabcBADadCBCbcDAdabCBACdcDadCDcabcBADadCBcbcDAd, abCBCbDAdacACADadBcbcDAd, abCBACdcDAdCDcDadCDADadcDAdabCBADadCDAdadcDAdCdcDadCDcabcBADadCBcbcDAd
(That is, four generators, three relators.)  The link complement has volume approximately 33.331392948.  Since the link is alternating and has no flypes, the diagram shown is minimal - thus the crossing number is 16.  One more thing - the diagram has an involution about the central vertical line.  You could mod out by this involution to get a smaller orbifold - there might be something known about the quotient...
A: Using appcontour I get the following Alexander polynomial:
$-u^6+4u^7-6u^8+4u^9-u^{10}+u^3v-7u^4v+23u^5v-42u^6v+47u^7v-33u^8v+13u^9v-2u^{10}v-4u^2v^2+24u^3v^2-66u^4v^2+104u^5v^2-106u^6v^2+73u^7v^2-29u^8v^2+5u^9v^2+5uv^3-29u^2v^3+73u^3v^3-106u^4v^3+104u^5v^3-66u^6v^3+24u^7v^3-4u^8v^3-2v^4+13uv^4-33u^2v^4+47u^3v^4-42u^4v^4+23u^5v^4-7u^6v^4+u^7v^4-v^5+4uv^5-6u^2v^5+4u^3v^5-u^4v^5$
Defined up to a change of base of $Z^2$
The second Alexander ideal is interesting, it is generated by
$<1133, v^2 - 531 v + 1, v - 10 u + 481 uv>$
(obtained using a combination of appcontour and Macaulay2)
The third Alexander ideal is the whole ring.
